cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A304444 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(2*n).

Original entry on oeis.org

1, 2, 14, 98, 726, 5512, 42614, 333608, 2636326, 20985272, 168012824, 1351507830, 10914317934, 88432329546, 718545161208, 5852747363518, 47774241056710, 390702055798978, 3200542803221192, 26257321971526646, 215705170816632376, 1774181109262878848
Offset: 0

Views

Author

Vaclav Kotesovec, May 12 2018

Keywords

Links

Crossrefs

Cf. A000712, A008485, A161870, A171803, A304443.

Programs

  • Mathematica
    nmax = 25; Table[SeriesCoefficient[Product[1/(1-x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[1/QPochhammer[x]^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{1/QPochhammer[r*s]^2 == s, 1/s + 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][r*s, r*s] == (2*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]))/(s* Log[r*s])}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2)/(Pi*(16*r*s*ArcTanh[1 - 2*r*s] - (-1 + r*s)*(Log[r*s] - 2*Log[1 - r*s])*(3*Log[r*s] - 2*Log[1 - r*s]) - 8*Log[1 - r*s] - 8*(-1 + r*s)*(-1 + 2*ArcTanh[1 - 2*r*s])* QPolyGamma[0, 1, r*s] + (4 - 4*r*s)* QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] + r*s*Log[r*s] * (r*s^(3/2)*Log[r*s]* Derivative[0, 2][QPochhammer][r*s, r*s] - 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ c * d^n / sqrt(n), where d = 8.42516721063251541777601555584151410936132980324698494327338254953123205... and c = 0.29923152009652750283923119244187982714171590056794904644563876...

A171802 G.f. satisfies: A(x) = P(x*A(x)^2) where A(x/P(x)^2) = P(x) is the g.f. for Partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 20, 115, 714, 4669, 31671, 220800, 1572395, 11389059, 83642650, 621400794, 4661706035, 35264616260, 268700873765, 2060348179869, 15886552304352, 123102352038195, 958128272163860, 7487015421267228, 58715989507106041
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 +...
G.f. satisfies A(x/P(x)^2) = P(x) where:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...
and x/P(x)^2 = x - 2*x^2 - x^3 + 2*x^4 + x^5 + 2*x^6 - 2*x^7 - 2*x^9 +...

Links

Crossrefs

Cf. A171803, A171804, A171805, A109085, A000041, A304444, A366026.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/Product[1 - x^k*A[x]^(2*k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{s*QPochhammer[r*s^2] == 1, 1/s + 2*r*s^2*Derivative[0, 1][QPochhammer][r*s^2, r*s^2] == (2*(Log[1 - r*s^2] + QPolyGamma[0, 1, r*s^2]))/(s* Log[r*s^2])}, {r, 1/8}, {s, 1}, WorkingPrecision -> 1000]; {N[1/r /. eq, 100], val = -s* Log[r*s^2]*(Sqrt[1 - r*s^2]/ Sqrt[4*Pi*(16*r*s^2*ArcTanh[1 - 2*r*s^2] + (1 - r*s^2)*(Log[r*s^2] - 2*Log[1 - r*s^2])*(3*Log[r*s^2] - 2*Log[1 - r*s^2]) - 8*Log[1 - r*s^2] + 8*(1 - r*s^2)*(-1 + 2*ArcTanh[1 - 2*r*s^2]) * QPolyGamma[0, 1, r*s^2] + 4*(1 - r*s^2)*QPolyGamma[0, 1, r*s^2]^2 - 4*(1 - r*s^2)*(QPolyGamma[1, 1, r*s^2] + r*s^2*Log[r*s^2]*(r*s^3*Log[r*s^2]* Derivative[0, 2][QPochhammer][r*s^2, r*s^2] - 2* Derivative[0, 0, 1][QPolyGamma][0, 1, r*s^2])))]) /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Sep 26 2023 *)
  • PARI
    a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^2))^(1/2), n)

Formula

G.f. A(x) satisfies [Paul D. Hanna, Nov 24 2012]:
(1) A(x) = (1/x)*series_reversion(x*eta(x)^2).
(2) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^(2*n)).
(3) A(x) = Sum_{n>=0} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k*A(x)^(2*k)).
(4) A(x) = Sum_{n>=0} (x*A(x)^2)^(n^2) / Product_{k=1..n} (1-x^k*A(x)^(2*k))^2.
(5) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^(2*n)/(1 - x^n*A(x)^(2*n)) ).
a(n) ~ c * d^n / n^(3/2), where d = 8.42516721063251541777601555584151410936... and c = 0.2128745515668564974075326286129891378270... - Vaclav Kotesovec, May 13 2018

A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).

Original entry on oeis.org

1, 3, 18, 130, 1044, 8946, 80135, 741312, 7027515, 67911855, 666525630, 6625647054, 66570488901, 674964968175, 6897258376218, 70961851119848, 734455079297433, 7641851681095236, 79886815507105175, 838655487787502616, 8837797224686207976, 93454820274339167191
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2009

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...
where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and
x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...

Crossrefs

Cf. A109085, A171802, A171803, A171804, A000041, A007312, A010815, A010816.

Programs

  • Mathematica
    InverseSeries[x QPochhammer[x]^3 + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^3 == s, 1/s + 3*(s/r)^(1/3)*Derivative[0, 1][QPochhammer][s, s] == (3*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/10}, {s, 1/8}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*(-1 + s)*s^2*(Log[s]^2/(6*Pi*(r*(-4*s*ArcTanh[1 - 2*s] + Log[1 - s]*(2 + 3*(-1 + s)*Log[1 - s] + Log[s] - s*Log[s])) - (-1 + s)*(-3*r*QPolyGamma[0, 1, s]^2 + r*QPolyGamma[1, 1, s] + QPolyGamma[0, 1, s]*(r*(2 - 6*Log[1 - s] + Log[s]) + 6*(r/s)^(2/3)*s^2*Log[s]* Derivative[0, 1][QPochhammer][s, s]) + s*Log[s]*((r/s)^(1/3)*s*(6*(r/s)^(1/3) * Log[1 - s] * Derivative[0, 1][QPochhammer][s, s] - 4*s*Log[s] * Derivative[0, 1][QPochhammer][s, s]^2 + (r/s)^(1/3)*s*Log[s]* Derivative[0, 2][QPochhammer][s, s]) - 2*r*Derivative[0, 0, 1][ QPolyGamma][0, 1, s])))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)
  • PARI
    {a(n)=polcoeff(serreverse(x*eta(x+x*O(x^n))^3),n)}

Formula

G.f. A(x) satisfies:
(1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;
(2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).
G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
Self-convolution cube of A171804 (with offset).
a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - Vaclav Kotesovec, Nov 11 2017

Extensions

More terms from Vladimir Reshetnikov, Nov 21 2016

A171804 G.f. satisfies: A(x) = P(x*A(x)^3) where A(x/P(x)^3) = P(x) is the g.f. for partition numbers (A000041).

Original entry on oeis.org

1, 1, 5, 33, 252, 2090, 18299, 166450, 1557595, 14898228, 145003996, 1431487820, 14299208690, 144262270360, 1467857359738, 15045486643137, 155208575698230, 1610201799670560, 16788969497000365, 175838914655128068
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2009

Keywords

Examples

			From _Peter Bala_, Nov 12 2024: (Start)
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + ...
I(P(x)) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + 2816*x^7 + ...
I^2(P(x)) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 + 4669*x^6 + 31671*x^7 + ...
I^3(P(x)) = 1 + x + 5*x^2 + 33*x^3 + 252*x^4 + 2090*x^5 + 18299*x^6 + 166450*x^7 + ... = the g.f. A(x). (End)

Crossrefs

Cf. A000041, A010815, A109085, A145268, A171802, A171803, A171805.

Programs

  • PARI
    a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^3))^(1/3), n)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A^3+x*O(x^n))^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(3*m)/prod(k=1, m, (1-x^k+x*O(x^n))))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(3*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^3+x*O(x^n))))); polcoeff(A, n)}

Formula

G.f. satisfies
(1) A(x) = 1/Product_{k>0} (1-x^k*A(x)^3).
(2) A(x) = Sum_{n>=0} x^n*A(x)^(3*n) / Product_{k=1..n} (1-x^k*A(x)^(3*k)).
(3) A(x) = Sum_{n>=0} x^(n^2)*A(x)^(3*n^2) / Product_{k=1..n} (1-x^k*A(x)^(3*k))^2.
G.f.: A(x) = 1 + x + 5*x^2 + 33*x^3 + 252*x^4 + 2090*x^5 + ...
G.f. satisfies A(x/P(x)^3) = P(x) where:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + ...
and x/P(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + ...
Also, the g.f. A = A(x) satisfies:
(1) A(x) = 1/((1-x*A^3) * (1-x^2*A^6) * (1-x^3*A^9) * (1-x^4*A^12) * ...).
(2) A(x) = 1 + x*A^3/(1-x*A^3) + x^2*A^6/((1-x*A^3)*(1-x^2*A^6)) + x^3*A^9/((1-x*A^3)*(1-x^2*A^6)*(1-x^3*A^9)) + ...
(3) A(x) = 1 + x*A^3/(1-x*A^3)^2 + x^4*A^12/((1-x*A^3)*(1-x^2*A^6))^2 + x^9*A^27/((1-x*A^3)*(1-x^2*A^6)*(1-x^3*A^9))^2 + ...
From Peter Bala, Nov 12 2024: (Start)
A(x) = ( 1/x * series_reversion(x/P(x)^3) )^(1/3).
A(x) = the third iterate I^3(P(x)), where the operator I is defined by I(f(x)) = 1/x * series_reversion(x/f(x)). See the Example section. (Note that I(P(x)) is the g.f. of A109085 and I^2(P(x)) is the g.f. of A171802.) (End)
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